Incompressible surfaces in 2-bridge knot complements
type: Item
instance of: scholarly article
title: Incompressible surfaces in 2-bridge knot complements
zbMATH Open document ID: 0602.57002
DOI: 10.1007/BF01388971
author: Allen Hatcher
author: William P. Thurston
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143193
review text: This paper gives the isotopy classification of incompressible and \(\partial\)-incompressible surfaces (orientable or not) in the complement of a 2-bridge knot \(K_{p/q}\). The only closed incompressible surfaces are boundary parallel tori. For each continued fraction expansion \(p/q=r+[b_ 1,...,b_ k]\) there is a branched surface \(\Sigma (b_ 1,...,b_ k)\) in \(S^ 3\) which contains all the surfaces obtained by plumbing together k bands in a row, the ith band having \(b_ i\) half- twists. (These surfaces all have boundary \(K_{p/q}.)\) All the incompressible \(\partial\)-incompressible surfaces are obtained by plumbing ''multiple bands''. The questions of which of these are connected and which are orientable are discussed, but not dealt with completely. The slopes of their boundary curves are given. The orientable incompressible Seifert surfaces for K all have the same genus, and are all isotopic if and only if in the unique continued fraction expansion of p/q with all \(b_ i's\) even at most one is not \(\pm 2.\) Except when \(K_{p/q}\) is a torus knot, Dehn surgery always gives an irreducible 3-manifold, and almost all such are hyperbolic but not Haken.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 3971768
zbMATH Keywords: non-Haken 3-manifolds
zbMATH Keywords: incompressible and \(\partial \)-incompressible surfaces
zbMATH Keywords: complement of a 2-bridge knot
zbMATH Keywords: Seifert surfaces
zbMATH Keywords: hyperbolic
reviewed by: Jonathan A. Hillman
MaRDI profile type: Publication
cites work: Incompressible Surfaces in Once-Punctured Torus Bundles
cites work: Bounded, separating, incompressible surfaces in knot manifolds
cites work: Incompressible surfaces in punctured-torus bundles
cites work: The Space of Incompressible Surfaces in a 2-Bridge Link Complement
cites work: Incompressible surfaces via branched surfaces
cites work: On the boundary curves of incompressible surfaces
cites work: Closed incompressible surfaces in alternating knot and link complements
cites work: Elementary surgery along a torus knot
cites work: Closed incompressible surfaces in complements of star links
cites work: Incompressible branched surfaces
cites work: Incompressibility of surfaces after Dehn surgery
cites work: Homotopieringe und Linsenräume
cites work: Parabolic Representations of Knot Groups, I
cites work: Knoten mit zwei Brücken
cites work: Elastic waves in rods and clad rods
cites work: Three dimensional manifolds, Kleinian groups and hyperbolic geometry
cites work: Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II
description: scientific article; zbMATH DE number 3971768